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Properties

Label 2-336-21.5-c3-0-18
Degree $2$
Conductor $336$
Sign $0.577 + 0.816i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.48 − 2.62i)3-s + (−5.27 + 9.13i)5-s + (−17.7 − 5.44i)7-s + (13.1 + 23.5i)9-s + (−26.6 + 15.4i)11-s + 19.8i·13-s + (47.6 − 27.0i)15-s + (−46.3 − 80.2i)17-s + (118. + 68.6i)19-s + (65.0 + 70.9i)21-s + (−37.6 − 21.7i)23-s + (6.89 + 11.9i)25-s + (2.87 − 140. i)27-s − 134. i·29-s + (−144. + 83.6i)31-s + ⋯
L(s)  = 1  + (−0.862 − 0.505i)3-s + (−0.471 + 0.816i)5-s + (−0.955 − 0.293i)7-s + (0.488 + 0.872i)9-s + (−0.731 + 0.422i)11-s + 0.423i·13-s + (0.820 − 0.466i)15-s + (−0.661 − 1.14i)17-s + (1.43 + 0.828i)19-s + (0.675 + 0.737i)21-s + (−0.341 − 0.197i)23-s + (0.0551 + 0.0956i)25-s + (0.0204 − 0.999i)27-s − 0.860i·29-s + (−0.839 + 0.484i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6955079021\)
\(L(\frac12)\) \(\approx\) \(0.6955079021\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.48 + 2.62i)T \)
7 \( 1 + (17.7 + 5.44i)T \)
good5 \( 1 + (5.27 - 9.13i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (26.6 - 15.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 19.8iT - 2.19e3T^{2} \)
17 \( 1 + (46.3 + 80.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-118. - 68.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (37.6 + 21.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 + (144. - 83.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-191. + 332. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 107.T + 6.89e4T^{2} \)
43 \( 1 - 285.T + 7.95e4T^{2} \)
47 \( 1 + (-120. + 209. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-432. + 249. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (366. + 634. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (265. + 153. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-280. - 485. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 74.2iT - 3.57e5T^{2} \)
73 \( 1 + (-141. + 81.6i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-437. + 757. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 406.T + 5.71e5T^{2} \)
89 \( 1 + (526. - 911. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 243. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.06811672837665672789345956065, −10.22927502506787049177149110629, −9.361594752777059223341730618088, −7.57604798390124415951236138542, −7.24277753705634798529378630916, −6.24338477623774713710454514675, −5.15133494435564157183205898499, −3.74658221831934678064062085509, −2.38968685749234472135832061742, −0.41452906118394966498057040471, 0.798507510516125537980852329949, 3.09501212379427608430080896957, 4.30352319136922467409539631451, 5.36804722836649124199970778536, 6.14885334309825715863797897693, 7.40179610906410808112792216062, 8.638576879614743159191934564485, 9.448928514268669020423842172390, 10.41184748227435715469992910536, 11.23928928107559924838748159385

Graph of the $Z$-function along the critical line